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Journal of
Plant Ecology
VOLUME 5, NUMBER 1,
PAGES 72–81
Coexistence of nearly neutral
species
MARCH 2012
doi: 10.1093/jpe/rtr040
available online at
www.jpe.oxfordjournals.org
Fangliang He1,2,*, Da-Yong Zhang3 and Kui Lin3
1
Department of Renewable Resources, University of Alberta, Edmonton, Alberta, Canada T6G 2H1
SYSU-Alberta Joint Lab for Biodiversity Conservation, State Key Laboratory of Biocontrol, Sun Yat-sen University,
Guangzhou 510275, China
3
State Key Laboratory of Earth Surface Processes and Resource Ecology and MOE Key Laboratory of Biodiversity Sciences and
Ecological Engineering, Beijing Normal University, Beijing, 100875 China
*Correspondence address. Department of Renewable Resources, University of Alberta, Edmonton, Alberta,
Canada T6G 2H1. E-mail: [email protected]
2
Aims
The neutral theory of biodiversity provides a powerful framework for
modeling macroecological patterns and interpreting species assemblages. However, there remain several unsolved problems, including
the effect of relaxing the assumption of strict neutrality to allow for
empirically observed variation in vital rates and the ‘problem of
time’—empirically measured coexistence times are much shorter
than the prediction of the strictly neutral drift model. Here, we develop a nearly neutral model that allows for differential birth and
death rates of species. This model provides an approach to study species coexistence away from strict neutrality.
Methods
Based on Moran’s neutral model, which assumes all species in a community have the same competitive ability and have identical birth
and death rates, we developed a model that includes birth–death
trade-off but excludes speciation. This model describes a wide range
of asymmetry from strictly neutral to nearly neutral to far from neutral
and is useful for analyzing the effect of drift on species coexistence.
Specifically, we analyzed the effects of the birth–death trade-off on
the time and probability of species coexistence and quantified the
loss of biodiversity (as measured by Simpson’s diversity) due to drift
by varying species birth and death rates.
INTRODUCTION
One of the most debated questions in the neutral theory of biodiversity is how long can neutral species coexist? Under the
assumption of strict neutrality, in which species in a community are assumed to have exactly the same vital rates, Hubbell
(2001) showed that species can coexist for a very long time,
long enough for high species diversity to be maintained by speciation alone. This coexistence is not maintained by the ability
of species to compete more effectively when rare but by the
Important Findings
We found (i) a birth–death trade-off operating as an equalizing force
driven by demographic stochasticity promotes the coexistence of
nearly neutral species. Species near demographic trade-offs (i.e. fitness equivalence) can coexist even longer than that predicted by the
strictly neutral model; (ii) the effect of birth rates on species coexistence is very similar to that of death rates, but their compensatory
effects are not completely symmetric; (iii) ecological drift over time
produces a march to fixation. Trade-off-based neutral communities
lose diversity more slowly than the strictly neutral community, while
non-neutral communities lose diversity much more rapidly; and (iv)
nearly neutral systems have substantially shorter time of coexistence
than that of neutral systems. This reduced time provides a promising
solution to the problem of time.
Keywords: birth rate d ecological drift d death rate d
demographic trade-off d nearly neutral theory d probability of
extinction d time of coexistence
Received: 17 July 2011 Revised: 7 October 2011 Accepted: 17
October 2011
stochasticity of ecological drift. The introduction of drift as
an explanation of species assemblages, as opposed to various
models of niche partitioning (Tokeshi 1999), is arguably
the most significant contribution of neutral theory. This contribution has, however, been incisively debated on several
fronts. Zhang and Lin (1997) first questioned the fragility of
the coexistence of neutral species by analytically showing that
a small departure from the assumption of equal birth rate
would dramatically reduce the time of coexistence. Along this
vein, Yu et al. (1998) demonstrated, by simulation, a similar
Ó The Author 2012. Published by Oxford University Press on behalf of the Institute of Botany, Chinese Academy of Sciences and the Botanical Society of China.
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Abstract
He et al.
|
Coexistence of nearly neutral species
(2012), our model is non-neutral by definition. It differs from
the previous models in two important aspects. First, unlike the
previous non-neutral models which only consider the asymmetry in either birth rates (Zhang and Lin 1997; Zhou and
Zhang 2008) or death rates (Fuentes 2004; Yu et al. 1998),
the simultaneous consideration of differential birth and death
rates allows us to investigate the effect of a demographic tradeoff on species coexistence away from the neutral state, which is
otherwise impossible to do. Second, unlike the
fitness-equivalence neutral models of Lin et al. (2009), Ostling
(2011) and Zhang et al. (2012), what our model considers is
a general case in which Bi/Di and Bj/Dj are not necessarily equal
but can vary independently. We use this model to examine the
joint effect of biased birth and death rates on the probability of
species extinction and the time of coexistence and show that
a demographic trade-off can equalize and contribute to species
coexistence without invoking stabilizing mechanisms. We
then investigate the effect of ecological drift on the loss of diversity under non-neutral conditions. We conclude the study
by discussing the consequences of the nearly neutral theory,
including the ‘problem of time’—the age of species predicted
by ecological drift is simply too long to be realistic (Lande et al.
2003; Nee 2005; Ricklefs 2006; Rosindell et al. 2010).
THE NEARLY NEUTRAL MODEL
Our departure point is Moran’s neutral model, assuming all
species in a community have the same competitive ability
and have identical birth and death rates (Moran 1962). Moran’s model is long standing, of which Hubbell’s zero-sum
model is a variant (Etienne and Alonso 2007; Leigh 2007). Because they differ little, we make no distinction between them.
Also, because the focus of this study is to investigate species
coexistence, to simplify the analysis, we only consider a system
consisting of two competing species, as many competition
models do. Under the assumption that the two species have
the same per capita birth and death rates and within a sufficiently small time step, the transition probabilities that the
abundance of the focal species increases by one, decreases
by one, or remains unchanged are, respectively (Hubbell
2001; Moran 1962):
pj;j + 1 =
N j j
N N
pj;j 1 =
j N j
N N
;
ð1Þ
pj;j = 1 pj;j + 1 pj;j 1
where N is the total number of individuals (the community
size), j is the number of individuals of the focal species (species
1) at time t and N j is the abundance of the other species
(species 2). The first term, Nj
N , on the right-hand side of pj,j+1
is the probability that a death occurs in species 2. The second
term, Nj , is the probability that a birth occurs in species 1. The
transition probability (pj,j+1) can therefore be interpreted as
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consequence for the violation of equal death rate. Subsequent
studies have extended the work of Zhang and Lin (1997) and
Yu et al. (1998) by investigating the effect of non-neutrality on
community diversity patterns (species-abundance and speciesarea curves). Fuentes (2004) first extended Yu et al.’s work
by simulating a spatially explicit nearly neutral system by considering slight differences in the viability of individuals. In
Fuentes’ simulation, the mortality rates of individuals are
no longer equal but are determined by small differential viabilities generated by deleterious versus beneficial mutation.
This nearly neutral community reduces species richness and
produces species-abundance and species-area curves very different from the strictly neutral model of Hubbell. Similar
results have also been reported by Zhou and Zhang (2008) that
simulated the effects of differential birth rates (as proposed by
Zhang and Lin 1997) on species-abundance distributions.
These models depart from Hubbell’s neutral model by only
considering species difference in either birth rate or death rate,
but not both, although in reality species differ in both rates.
The effect of a birth–death trade-off on species coexistence
has not yet been addressed.
As a significant step toward this problem, based on the lottery
model of demographic trade-off, Lin et al. (2009) showed that
although both a strictly neutral community and trade-off-based
communities produce log-series species-abundance distributions, species richness is higher in trade-off communities. Similarly, Ostling (2011) and Zhang et al. (2012) relaxed Hubbell’s
original neutral model by considering fitness equivalence and
found that high species diversity can result from an interspecific
birth–death trade-off. Parsons and Quince (2007) considered
a density-dependent ‘quasi-neutral’ model and analyzed the effect of population fluctuation on species coexistence. In these
studies, fitness equivalence is defined as Bi/Di = Bj/Dj = c, where
B and D are the birth and death rates for species i and j. Therefore, species having higher birth and death rates are considered
ecologically equivalent to species of lower birth and death rates.
Unlike Zhang and Lin (1997) and Yu et al. (1998), the trade-offbased models of Lin et al. (2009), Ostling (2011) and Zhang et al.
(2012) are still neutral by definition (Hubbell 2005), and the
strictly neutral model is the special case of Bi = Bj and Di = Dj.
Inspired by the theory of nearly neutral alleles in population
genetics that extends the neutral theory of molecular evolution (Ohta 1992; Ohta and Gillespie 1996), in this study, we
propose a nearly neutral model that describes a spectrum of
communities ranging from strictly neutral to non-neutral.
The model is derived based on two seminal works in population genetics: Moran (1962) and Karlin and McGregor (1967),
and it takes account of differential birth and death rates. Although our model can describe a wide range of asymmetry
(from strictly neutral to nearly neutral to non-neutral), it will
become clear that the model behaviors are very similar at
nearly neutral and far from neutral states. We therefore call
it a nearly neutral model.
Similar to Zhang and Lin (1997) and Yu et al. (1998) but different from Lin et al. (2009), Ostling (2011) and Zhang et al.
73
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Journal of Plant Ecology
a joint probability of two events: first sample one individual
from species 2 (to die) and then sample a second individual
from species 1 (to reproduce). The random walk of model 1
describes the abundance dynamics of species with same competitive ability—the probabilities of moving one step to the
Pj;j+1
right or to the left are equal: Pj;j1
=1.
Model 1 can be generalized by relaxing the assumption of
neutrality. This generalized model takes account of differential
birth and death rates so that asymmetry in the theory is
allowed. Adopting the approach proposed by Karlin and
McGregor (1967) to model 1, we obtain an asymmetric model:
pj; j + 1 =
N j
bj
N j + dj N j + bj
dj
N j
N j + dj N j +; bj
N 1
+ qi
i=j
ej =
i
where qi = qp11 qp22 ...q
...pi , 0 < i < N. pi and qi are the transition probabilities, defined by pi = pi;i+1 and qi = pi;i1 . The extinction probability can be analytically derived by substituting Equation (2) into (3):
(
ð2Þ
ej =
Pj; j = 1 pj;j + 1 pj;j 1
where the differential birth and death factors are, respectively,
denoted by b and d. When b = 1, the two species have the same
birth rate (B1 = B2); b > 1 means that the focal species has
a higher birth rate than species 2 (B1 > B2) and b < 1 means that
species 1 has a lower birth rate. Similar interpretations apply to
death factor, d. Note that when d = 1 and b 6¼ 1, Equation (2)
reduces to Zhang and Lin’s model (1997), when b = 1 and d 6¼ 1,
it is equivalent to Yu et al.’s (1998) model and when d = 1 and
b = 1, it further becomes Moran–Hubbell’s zero-sum model 1.
The transition probabilities in Equation (2) are easy to derive
(Karlin and McGregor 1967). They are arrived by considering
birth and death as sampling events. Let us take the two terms
in pj,j+1 as an example. Because of the species-biased death
rates, we denote D1 as the probability a death occurs in species
1 (with abundance = j) and D2 as the probability a death occurs
in species 2 (with abundance = N j). Because the death must
either occur in species 1 or 2, D1 + D2 = 1 (D1 = D2 for the neutral model). If we randomly sample one individual from the
community (to die), the probability that the sampled individNj
2 ðNjÞ
ual is species 2 is D1Dj+D
. This leads to Nj+dj
, where d=DD12 . To
2 ðNjÞ
derive the second term, we denote B1 as the probability of sampling an individual from species 1 and B2 the probability of
sampling an individual from species 2. If we randomly sample
one individual from the community (to give birth), then the
probability that the sampled individual belongs to species 1
j
bj
is B1 j+BB21ðNjÞ
, leading to Nj+bj
, where b = BB12 . All the other terms
in Equation (2) can be derived in the same manner. A salient
feature of this nearly neutral model is that drift in abundance is
no longer neutral but is biased, determined by the ratio of
Pj;j+1
b and d: Pj;j1
= db, which is considered as a measure of fitness inequality (sensu Chesson 2000). Species 1 and 2 are said to have
identical fitnesses if d = b, which is model 1.
What Equation (2) differs from Equation (1) is that the chances to walk to the right and to the left are no longer equal but
determined by the difference in species competitive ability (as
defined by b and d). Competitive exclusion in the system occurs
if species 1 starting from j is finally absorbed into state 0 (species
ð3Þ
; 0 < j < N;
N 1
1 + + qi
i=1
h
i
ðd=bÞj 1 ðd=bÞN j
1 ðd=bÞN
N j
N
b 6¼ d:
ð4Þ
b=d
Note that in Lin et al. (2009), only the probabilities for
b = d are considered (e.g. see Fig. 1 of Lin et al. 2009). This
study generalizes the previous results by considering any ratio
of b/d.
The second function is the time to absorption. This time can
be measured either by relative time or absolute time. The relative time is defined as the average number of steps for the
focal species traveling from j to either 0 or N. It can be derived
following the procedure of Taylor and Karlin (1998):
N 1
+ Ri
j1
i=1
tj = 1 + + qi
+ Ri ; 1<j<N;
ð5Þ
N 1
i=1
i=1
1 + + qi
i=1
qi
qi qi1
1
2
where Ri = pi 1+ pi1 + pi1 pi2 + + pqi1i qpi1i2...q
Substituting
...p1 :
Equation (2) into (5), we obtain
(
tj =
=
j1
1
1 d=b
"
N i 1 ðd=bÞj N 1 1
d
1
N +
b
1 ðd=bÞ i = 1 pi
j i #
j1
1
d
1
+
b 6¼ d
b
i = 1 pi
=
j1
j N1 N i
ji
+
+
N i = 1 pi
i = 1 pi
ð6Þ
b=d
The absolute time to extinction is the actual time for the focal
species traveling from j to either 0 or N state. The derivation of
this absolute time requires reformulation of Equation (2) and can
only be done for the case of fitness equivalence, as given in
Appendix 1. Because relative time is most commonly used in
the literature (Ewens 2004; Hubbell 2001; Zhang and
Lin 1997) and is also more useful to present the concept of nearly
neutral theory, in the rest of this study, relative time is used.
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pj; j 1 =
2 wins) or N (1 wins). We use two functions to investigate the
effects of a birth–death trade-off on the outcome of competition.
The first function is the probability of absorption, defined as
the probability that the focal species starting from j will go extinct (Taylor and Karlin 1998):
He et al.
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Coexistence of nearly neutral species
75
Ecological drift is the only force regulating species coexistence in models 1 and 2. It causes the loss of species and over
time produces a march to fixation. A classical model of population genetics is that genetic drift drives the loss of alleles at
t
the rate of Ht = H0 1 N1 , where Ht is the heterozygosity in
population generics at time t and N is population size. This
model can be generalized to quantify the effect of ecological
drift on species diversity. Considering a lottery model of strictly
neutral species (i.e. all species have the same annual death rate
D and the same annual birth rate B), we derived the effect of
ecological drift on the loss of species diversity as (Appendix 2)
t
2D D2
Ht = H0 1 ;
N
ð7Þ
where Ht is the Simpson diversity index (i.e. heterozygosity) at
time t. It is clear that under ecological drift, communities are
bound to fixation over time, resulting in HN = 0. Note that
Equation (7) is a general model that applies to communities
of multiple species, not just two species.
We conducted simulations based on the lottery model to evaluate the effect of drift on non-neutral communities by varying
species birth and death rates. The lottery model describes the
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Figure 1: effects of different birth (b) and death (d) rates on the extinction probability of the focal species for three initial states j = 10 (a, b),
50 (c, d) and 90 (e, f) for model 2. The right-hand panels are the contour plots of their left-hand 3D plots.
76
Journal of Plant Ecology
the strictly neutral around b d > 1 (along the red lines of Fig.
2b, d and f) that represents a neutral model of approximate
fitness equivalence. Note coexistence time is the same, as long
as b = d since tj in Equation (6) only depends on d/b.
The effect of ecological drift on the loss of biodiversity varies
between neutral and non-neutral communities (Fig. 3).
Simpson’s diversity for the simulated strictly neutral community (B/D = 1; Fig. 3a) is well predicted by the theoretical
model (Eq. 7). However, demographic differentiation plays
a sharply contrasting role in maintaining diversity, depending
on whether there is trade-off in demographic difference or
not. For trade-off communities, the loss of diversity is slower
than the strictly neutral community (Fig. 3b), while for nontrade-off communities, the loss of Simpson’s diversity is accelerated over time (Fig. 3c and d). These results are in agreement with the finding that demographic trade-off prolongs
coexistence time (Fig. 2).
RESULTS
DISCUSSION
We analyzed the combined effects of biased birth (b) and death
(d) rates on ej and tj. In the following evaluations, N = 100 was
used. Larger N can be used, but the summation term involving
ðd=bÞx in the first equation of Equation (6) causes numerical
overflow. N = 8 or 16 are commonly used in numerical evaluations of Hubbell (2001) due to the computation problem. Experimentation with other N did not change the qualitative results.
To evaluate the effect of initial state j on ej and tj, we considered three initial abundances from which the focal species
started: j = 10, 50 and 90 (corresponding to 10, 50 and 90% of
N). In the evaluations, the ranges of b and d were varied from
1 to 1.5. Higher b (d) means the focal species has higher birth
(death) rates than species 2.
The effects of b and d on the extinction probability (ej) of the
focal species are shown in Fig. 1. For small initial j, ej is high
(Fig. 1a). This trend is reversed with the increase of j (Fig. 1c
and e). In all of the three cases, extinction increases with d but
decreases with b. There is a clear trade-off between b and d: the
adverse effect of d is compensated for by b.
Unlike the probability of extinction, which describes the extinction of the focal species, the time of coexistence as defined
by Equation (6) is the time for which the system wanders before a species (1 or 2) hits an absorption state (0 or N). The
effects of b and d on the time of coexistence are shown in
Fig. 2. Across j, the coexistence time is highest when j = 50
(half of the community size). With each j, for a fixed b, the
time increases with d to a peak and then decreases (Fig. 2). This
trajectory also applies to the effect of b by fixing d. The peak
ridges shown in Fig. 2 are maintained by the trade-off of
b and d—a high birth rate b is offset by a high death rate d.
However, the trade-offs are not symmetric—b needs to be
slightly higher than d to maintain the ridge when j < N/2
(Fig. 2b) or d slightly higher than b (Fig. 2f). An interesting
result is that the highest coexistence time is not maintained
at b = d = 1 (strictly neutral state) but at the state away from
The debate over the neutral theory of biodiversity is ongoing
(Clark 2009; Rosindell et al. 2011). Among the several unsolved problems, the strongest criticism of the theory is its unrealistic assumption of perfect neutrality, which ignores
observed difference in species fitness. Attempts have been
made to reconcile the neutral and niche theories and to generalize neutral theory by incorporating asymmetry (Adler et al.
2007; Allouche and Kadmon (2009); Alonso et al. 2006, 2008;
Cadotte 2007; Gravel et al. 2006; Hubbell 2005; Leibold and
McPeek 2006; Ostling 2011; Peng et al. 2012; Rosindell et al.
2010, 2012; Tilman 2004; Volkov et al. 2005; Zhang et al.
2012; Zillio and Condit 2007). Our study is a useful step along
this direction. We term this theory as nearly neutral theory
sensu the nearly neutral theory of molecular evolution (Ohta
1992; Ohta and Gillespie 1996). It is nearly neutral because our
results (Figs 1 and 2) show that a small perturbation to the
symmetry assumption of birth and death rates dramatically
reduces species coexistence; the effect decelerates as the differences in birth and death rates increase.
Equation (2) is a generalization of previous models that
take account of demographic differentiation. It becomes
a strictly neutral model when b = d = 1. The increase or decrease of b and d describes departure from the neutral model
and reflects differences in species fitness. Neutral-dominated
communities will have b and d near 1, while non-neutral
communities have b and d larger than 1. The fitness equivalent models of Lin et al. (2009), Ostling (2011) and Zhang et al.
(2011) correspond to the special case of b = d 6¼ 1. It is obvious
from Equations (4) to (6) that the extinction probability and
coexistence time not only depend on the d/b ratio but also on
the initial size of the focal species. This makes the behavior of
the fitness equivalent models be the same as the strict neutral
community of Equation (1). When b 6¼ d, Equation (2) corresponds to the non-neutral models of Zhang and Lin (1997)
and Yu et al. (1998), and the analysis of Yu et al. (1998) is
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same drift process of model 2 but is easier to program when dealing with multiple species (Lin et al. 2009). The detail of the simulation algorithm is described in Appendix 3. To sketch the
algorithm, we started off with communities of 100 equally abundant species with community size = 10 000 individuals. Thus,
H0 = 100/101 (calculated from sampling without replacement).
To generate the non-neutral communities, for each species, we
first chose a death rate D from a uniform distribution, U(0, 1).
The birth rate B for that species is then defined as B ; N(D,
r2) and B > 0. Note that here, without loss of generality, the birth
rates of species are scaled to lie also between 0 and 1. In this
study, we present results for four scenarios with r = 0.01,
0.001, 0 (trade-off communities) and a strictly neutral configuration with the death rate D = B = 0.5 for each species, since each
simulated community has an average death rate of 0.5 at start. In
total, the simulation took 100 runs. For each run, the observation
time is 60 000 birth–death cycles.
He et al.
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Coexistence of nearly neutral species
equivalent to that of Zhang and Lin (1997). This finding is
previously unrecognized and reveals that the effect of differential death on species coexistence is very similar to the effect
of differential birth.
However, it is important to note two points. The first is that
the compensatory effects of birth and death on coexistence are
not completely symmetric as shown by Fig. 2 (if the effect were
77
completely symmetric, the slope of the red line in Fig. 2d
would be 1 with intercept = 0). The second point is that the simultaneous consideration of birth and death in the nearly neutral model has greatly generalized our understanding of the
effect of demographic trade-off on coexistence. Previous results
(Fuentes 2004; Yu et al. 1998; Zhang and Lin 1997; Zhou and
Zhang 2008), by only considering birth differentiation or death
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Figure 2: effects of different birth (b) and death (d) rates on the time of coexistence for three initial states j = 10 (a, b), 50 (c, d) and 90 (e, f) for
model 2. The right-hand panels are the contour plots of their left-hand 3D plots. The peak ridge of the coexistence time is shown by the red line in
each contour plot: d = 0.0468 + 0.909b, d = 0.0315 + 0.971b and d = 0.05 + b, respectively, for panels b, d and f.
78
Journal of Plant Ecology
differentiation, just correspond to slice profiles of Figs 1 and 2 by
fixed b or d and thus could not unveil the effect of demographic
trade-off. Furthermore, the consideration of both birth and
death is necessary if other processes (e.g. density dependence)
in addition to the neutral process are considered (Parsons and
Quince 2007).
Species of the strictly neutral model have identical fitness,
and their dynamics are solely driven by random drift. This
system does not involve non-linearity to ensure rare species
advantage (Chesson and Warner 1981). By this criterion, the
nearly neutral model is also an unstable model that does not
involve stabilizing (rare species advantage) but equalizing
mechanisms (demographic stochasticity) that maintain coexistence. Figure 2 shows that coexistence time is higher
at states away from the strictly neutral along the fitnessequivalence manifold. This occurs due to a strong birth–
death trade-off between b and d by which symmetry (b d 6¼ 1 but not b = d = 1) is nearly met (Fig. 2b, d and f). In
this region, species coexist for a very long time, even longer
than strictly neutral species. This result is consistent with the
finding of Lin et al. (2009), Ostling (2011) and Zhang et al.
(2011) that higher diversity can be maintained by neutral
model of equivalent fitness than by the strictly neutral model.
The demographic trade-off here operates as an equalizing force
that tends to minimize fitness difference between species. This
result provides a support to Hubbell’s claim that interspecific
differentiation can promote coexistence without invoking
niche mechanisms (stabilizing force). It is worth noting that
trade-offs in life history traits are typically regarded as a niche
stabilizing mechanism (Grubb 1977). As we have shown here
this is not necessarily true as far as demographic trade-off is
concerned.
Ecological drift, dispersal limitation and speciation are the
three major community assembly rules in Hubbell’s neutral
theory. However, little is currently understood on how drift
would reduce the diversity of communities. For example,
how should effective community size (analogous to effective
population size in population genetics) be defined to measure
the loss of diversity due to drift (Hu et al. 2006; Lande et al.
2003; Orrock and Fletcher 2005)? Results in Fig. 3 show that
fitness equivalent neutral communities (Fig. 3b) lose diversity
more slowly than the strictly neutral community, while nearly
neutral communities (Fig. 3c and d) lose diversity much more
rapidly than the strictly neutral community. Clearly, the effective community size for nearly neutral communities should be
smaller than the neutral community. The challenge is how we
may analytically define an effective community size that
incorporates b and d (and also immigration and speciation processes) (Hu et al. 2009).
The strictly neutral theory has been criticized for failing to
correctly predict the time of species existence (Lande et al.
2003; Nee 2005; Ricklefs 2006). Theories in population genetics establish that the average age of a population in a community is approximately 2N (N is community size) (Kimura
1983; Leigh 1981). This leads to the prediction that the average age of a tree species, even if rare, in a region of size of
Central America would be billions of years, long before plants
originated (Nee 2005). Therefore, the process of drift is just
too slow to account for rapid species turnover over time.
There have been attempts to explain this problem. For
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Figure 3: effect of ecological drift on the loss of diversity as measured by Simpson’s index (heterozygosity). Red curves: the theoretical prediction
with a death rate of 0.5 for each species (Eq. 7); blue curves: mean 6 standard error; green curves: median. The simulated communities are strictly
neutral (a), trade-off (b), and non-neutral (c for r = 0.001 and d for r = 0.01).
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Coexistence of nearly neutral species
APPENDIX 1
Derivation of the absolute coexistence time for the
trade-off-based drift model
The derivation of the absolute coexistence time requires
a reformulation of the asymmetric Moran–Hubbell model 2
in terms of a non-linear one-step stochastic process. Let’s start
by assuming that the probability for a given individual of the
focal species to die in the time interval (t, t + Dt) is D1Dt, where
D1 is a species-specific ‘death rate’. From this assumption, it
follows that the probability of one death occurring in (t,
t + Dt) in the focal species is jD1 Dt+oðDtÞ, where oðDtÞ is a higher
order term of Dt and can be disregarded when Dt is small. The
above probabilities then is jD1 Dt.
The one-step transition probabilities that the abundance of
the focal species increases or reduces by one individual, or unchanged are, respectively:
B1 j
B1 j + B2 ðN jÞ
B2 ðN jÞ
;
pj; j 1 = ½jD1 Dt
B1 j + B2 ðN jÞ
pj; j + 1 = ½ðN jÞD2 Dt
pj; j = 1 pj;j + 1 pj;j 1
where Bi is the birth rate of species i (=1, 2). The first term,
½ðN jÞD2 Dt, on the right-hand side of pj,j+1 is the probability
of a death occurring in species 2 during the interval (t,
t + Dt). The second term, B1 j+BjB2 1ðNjÞ, is the probability of a birth
(immediately following the death event) occurring in species 1.
The transition probability, pj,j+1, is a joint probability of a death in
species 2 followed by a birth in species 1. The probabilities above
are the continuous time version of Equation (2) in the main text.
The time to fixation or absorption, tj, is defined as the average number of time steps for the focal species traveling from
j to either N or 0 (Ewens 2004; Zhang and Lin 1997):
j1 i
j N 1 i
1
1
+ +
+ +
N i = 1 k = 1 pk;k + 1
i = 1 k = 1 pk;k + 1
j1 i
j N 1 i B1 k + B2 ðN kÞ
B1 k + B2 ðN kÞ
=
+ +
; 1<j<N:
+ +
N i = 1 k = 1 ðN kÞkB1 D2 Dt
i = 1 k = 1 ðN kÞkB1 D2 Dt
tj =
FUNDING
National Natural Science Foundation of China (D.Y.Z. and
K.L.); Ministry of Science and Technology of China (D.Y.Z.
and K.L.); Geomatics for Informed Decisions Network of
Canada (F.H.); Natural Sciences and Engineering Research
Council of Canada (F.L.); University of Alberta International
(the China-UofA Joint Research Lab program).
ACKNOWLEDGEMENTS
We thank Douglas Yu for their constructive comments, which have
much improved this study.
Conflict of interest statement. None declared.
ðA1Þ
ðA2Þ
From (A2), it is obvious that a natural measure of coexistence time is given by
j1 i
j N 1 i B1 k + B2 ðN kÞ
B1 k + B2 ðN kÞ
+ +
+ +
N i = 1 k = 1 ðN kÞkB1 D2
i = 1 k = 1 ðN kÞkB1 D2
j1 jðN 1Þ j N 1 B2 N i
ji
B2 j i
=
+
+
+
+
ND2
N i = 1 B1 D2 i
i
i = 1 D2 ðN iÞ B1 D2
Tj = tj Dt =
1<j<N:
ðA3Þ
The above absolute coexistence time Tj can be further simplified for fitness equivalent neutral communities by assuming
equivalent fitness, i.e. B1/D1 = B2/D2. This fitness equivalence
Downloaded from http://jpe.oxfordjournals.org/ at Sun Yat-Sen University on December 19, 2013
example, it has been hypothesized that the division of a large
ecosystem into smaller homogeneous neutral communities,
climate change, coevolutionary interactions and gradual
(rather than instantaneous) speciation within ecological systems may each contribute to reducing the time of coexistence
(Lande et al. 2003; Nee 2005; Ricklefs 2006; Rosindell et al.
2010). Our results show that a small relaxation from the neutral assumption can substantially reduce the time of coexistence (Fig. 2). Although this reduction may still not be large
enough to fully account for the problem, nearly neutral theory coupling with other processes (e.g. division of a large system, reduced effective community size, gradual speciation)
can offer a promising solution.
It is worth mentioning that although Equation (2) is in essence a demographic model, it can also be used to model the
effect of environmental stochasticity on species coexistence.
Birth and death factors b and d in Equation (2) are the ratios
of the probabilities of sampling species 1 over species 2 (to die
or to reproduce). Environmental factors that affect birth and
death could thus be incorporated into the nearly neutral models through b and d, and, in turn, their effects on the transition
probabilities can be modeled. This may be an opportunity to
introduce stabilizing mechanisms into the unstable nearly
neutral model.
The nearly neutral theory developed here generalizes the
neutral theory while not compromising the analytical tractability of the theory. The theory provides a promise to reconcile
mismatching predictions of the current neutral theory. Here
we would like to raise an overlooked issue in the current study.
In fitness equivalent models, species of high birth and high
death are considered ecologically equivalent to species of
low birth and low death. This is true for species coexistence
as measured by Equations (4) and (6) but may not be so with
regard to other community properties, especially, in response
to environmental stochasticity. The high birth–high death versus low birth–low death trade-off will induce a difference in
variance along the trade-off manifold that could have important implications to population survival and thus community
organization. This is not a straightforward question but
deserves a closer attention.
79
80
Journal of Plant Ecology
(corresponding to the scenario of b = d of Eq. 6) removes B1 and
B2 from (A3).
The above model becomes the Moran–Hubbell model if the
1
time step is defined as Dt = jD1 +ðNjÞD
. This is the implicit assump2
1
1
Ft = 1 ð1 DÞ2
+ 1
Ft 1 + ð1 DÞ2 Ft 1 :
N
N
tion behind the Moran model and all subsequent Markov-chain
After some rearrangement, we obtain
2D D2
:
1
1 Ft = 1 Ft 1
N
models in discrete times. Here, Dt is no longer invariant but varies
with relative species abundances of the community if species differ in their death rates. This property, which makes the transition
probabilities no longer suitable for use in the master equation,
has passed practically unnoticed in the literature. Of course, this
is not a problem if only demographically symmetric species are
quired for the focal species to drift to 0 or N. Substituting
1
into the relative time of Equation (A2), one
Dt = jD1 +ðNjÞD
2
obtains Equation (6) of the main text. The relative time
and absolute time are qualitatively similar in most cases except that when death rate is very low in which the absolute
time can be longer than the relative time. In all cases, both
times show that strictly neutral species (i.e. having exactly
the same birth and death rates) in general do not constitute
maximum coexistence, particularly when viewed from the
relative time. This is an important result of this study, new
to the literature.
APPENDIX 2
Derivation of diversity decay for strictly neutral
species with identical vital rates in a lottery model
Define Ft as the probability that two randomly sampled individuals at generation t are of the same species. 1 Ft is thus
Simpson’s diversity index, commonly denoted by Ht in population genetics. In the lottery model (Lin et al. 2009), a portion
D of individuals die and are replaced by offspring of all individuals with equal probability. When two individuals are sampled
at random, there is a probability of ð1 DÞ2 that neither of
these two is the new individual and the probability that at least
one is the new individual is 1 ð1 DÞ2 . If neither of the sampled individuals is new, then the probability they are of same
species is simply Ft1. If two sampled individuals are both new,
the probability that they are the progeny of the same individual in the previous generation is 1/N; if they are the progeny of
different individuals, which occurs with probability (1 1/N),
then the probability they are of the same species is Ft1. Putting
together, the probability two sampled individuals are of
the same species is N1 + 1 N1 Ft1 when they are both
new. By the same argument we can derive the same result
when only one sampled individual is new. We thus have
the recursion
ðA3Þ
Therefore, Simpson’s diversity index changes with time as
t
2D D2
:
Ht = 1 Ft = H0 1 N
ðA3Þ
Obviously, if D = 1, it reduces to the Wright–Fisher model
t
with non-overlapping generations, which is Ht = H0 1 N1 .
The larger D, the quicker the diversity decays. This must be
the case because higher annual mortality rates imply more
turnovers, which in turn leads to more effective drift.
APPENDIX 3
Simulation based on the lottery model for evaluating
the effect of ecological drift on non-neutral
communities
We performed the simulation by running 100 independent
communities. For each of the 100 communities, its size is
set to be 10 000. At the beginning of each simulation, the community is occupied by 100 species with equal abundance, and
each species is characterized with a death rate (Di) randomly
drawn from a uniform distribution U(0, 1) and a birth rate (Bi)
randomly drawn from a normal distribution N(Di, r2), where
r2 is the variance and Bi > 0.
With the above general setting, each simulation goes forward as follow.
STEP 1 (death process):
For each species, its abundance is denoted as Ni.
For each individual of the species, a number is randomly
generated from a uniform distribution U(0, 1).
If the random number generated is less than or equal to its
death rate (Di), the individual is designated as dead, and
the abundance of the species (Ni) decreases by one.
Thus, the total number of dead individuals across all existent species in the community could be counted and is
denoted as tNum.
STEP 2 (birth process):
For each of these tNum vacant sites released from the
deaths of STEP 1, a new individual is recruited, which
is sampled from one of the existent species in the community (before the death process) in accordance to its birth
rate (Bi) and its abundance (Ni).
One cycle (from death to birth process) of the simulation
is counted as one time unit. Each simulation continues till
the total time unit reaches 60 000. During the course of
Downloaded from http://jpe.oxfordjournals.org/ at Sun Yat-Sen University on December 19, 2013
considered. This perhaps explains why it was not noticed before.
In the Moran–Hubbell model, the relative coexistence time
is measured in terms of the average number of deaths re-
ðA3Þ
He et al.
|
Coexistence of nearly neutral species
simulation, the community was sampled once per 1000
time units (hence there are 60 Simpson’s indices for each
simulation).
The above STEP 1 and STEP 2 were repeated for 100 communities (i.e. 100 independent runs). The resultant curves
(Fig. 3) and their error bars were calculated based on mean
and standard error of these 100 communities.
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